Calculate the Phi Coefficient

Problem

Calculate the Phi coefficient (Matthews correlation coefficient for 2x2) between two binary variables. The Phi coefficient measures the association between two binary variables and ranges from -1 to 1.

Solution

def phi_coefficient(x: list[int], y: list[int]) -> float:
    n11 = sum(1 for a, b in zip(x, y) if a == 1 and b == 1)
    n10 = sum(1 for a, b in zip(x, y) if a == 1 and b == 0)
    n01 = sum(1 for a, b in zip(x, y) if a == 0 and b == 1)
    n00 = sum(1 for a, b in zip(x, y) if a == 0 and b == 0)

    numerator = (n11 * n00) - (n10 * n01)
    denominator = ((n11 + n10) * (n11 + n01) * (n00 + n10) * (n00 + n01)) ** 0.5

    if denominator == 0:
        return 0.0
    return round(numerator / denominator, 4)

Explanation

1. Build the 2x2 contingency table: count co-occurrences n11, n10, n01, n00.
2. Numerator: (n11 * n00) - (n10 * n01) captures the difference between concordant and discordant pairs.
3. Denominator: The geometric mean of the four marginal products normalizes the coefficient to [-1, 1].
4. A value of +1 indicates perfect agreement, -1 indicates perfect disagreement, and 0 indicates no association.

Complexity

• Time: O(n) where n is the number of observations
• Space: O(1)